Normal Factor Graphs and Holographic Transformations

TL;DR

This paper introduces holographic transformations for normal factor graphs, establishing a generalized Holant theorem that unifies holographic algorithms and duality principles, and formalizes a linear algebraic semantics for these graphs.

Contribution

It presents a novel framework connecting holographic transformations with normal factor graphs, generalizing Holant theorem, and formalizing their linear algebraic properties.

Findings

Unified holographic transformations and normal factor graphs

Generalized Holant theorem linking these concepts

New linear algebraic semantics for normal factor graphs

Abstract

This paper stands at the intersection of two distinct lines of research. One line is "holographic algorithms," a powerful approach introduced by Valiant for solving various counting problems in computer science; the other is "normal factor graphs," an elegant framework proposed by Forney for representing codes defined on graphs. We introduce the notion of holographic transformations for normal factor graphs, and establish a very general theorem, called the generalized Holant theorem, which relates a normal factor graph to its holographic transformation. We show that the generalized Holant theorem on the one hand underlies the principle of holographic algorithms, and on the other hand reduces to a general duality theorem for normal factor graphs, a special case of which was first proved by Forney. In the course of our development, we formalize a new semantics for normal factor graphs, which highlights various linear algebraic properties that potentially enable the use of normal factor graphs as a linear algebraic tool.